SELBERG TRACE FORMULA 15
discrete series; and can be made to vanish by taking an appropriate kind of
imaginary part, to be denoted by Jm (see [Z2], § 2(11); the notation "7m" is
not used there, however). Formula (1-3) then follows from (1.6). Finally,
(1.2) is a straightforward consequence of the spectral decomposition.
Two main applications of the trace formulae are generalized Veyl laws and
prime geodesic theorems. The Veyl laws for the compact case state
([Z3], § 5):
(1.7) Nr(r,T) = E (Op(r)u., u.) « T/ln T
kj|T 3 J (s,m)
(where (s,m) are as usual the (fl,W) parameters of a).
The prime geodesic theorems for the compact case state ([Z2], § 4):
(1.8) *r(«r,T) ^
= JLop^u,, u ^ ^ e ( K T
Here 7v,s,mare certain explicit constants ([Z2], § 3) which will be described
further in § 6.
The Veyl laws (1.7) are required in the proof of the prime geodesic
theorems (1.8). Also required is an estimate in the absolute sums
(1.9) |Nr(«r,T)| = S |0p(r)u-, u-|
|rj|T J J
The obvious estimate